Pub Date : 2025-11-18DOI: 10.1016/j.difgeo.2025.102310
Tomasz Goliński , Alice Barbora Tumpach
In this paper, we investigate the theory of R-brackets, Baxter brackets and Nijenhuis brackets in the Banach setting, in particular in relation with Banach Poisson–Lie groups. The notion of Banach Lie–Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of a non-degenerate invariant pairing on a Banach Lie algebra, these equations of motion assume a Lax form. We prove a version of the Adler–Kostant–Symes theorem adapted to R-matrices on infinite-dimensional Banach algebras. Applications to the resolution of Lax equations associated to some Banach Manin triples are given. The semi-infinite Toda lattice is also presented as an example of this approach.
{"title":"Banach Poisson–Lie groups, Lax equations and the AKS theorem in infinite dimensions","authors":"Tomasz Goliński , Alice Barbora Tumpach","doi":"10.1016/j.difgeo.2025.102310","DOIUrl":"10.1016/j.difgeo.2025.102310","url":null,"abstract":"<div><div>In this paper, we investigate the theory of <em>R</em>-brackets, Baxter brackets and Nijenhuis brackets in the Banach setting, in particular in relation with Banach Poisson–Lie groups. The notion of Banach Lie–Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of a non-degenerate invariant pairing on a Banach Lie algebra, these equations of motion assume a Lax form. We prove a version of the Adler–Kostant–Symes theorem adapted to <em>R</em>-matrices on infinite-dimensional Banach algebras. Applications to the resolution of Lax equations associated to some Banach Manin triples are given. The semi-infinite Toda lattice is also presented as an example of this approach.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102310"},"PeriodicalIF":0.7,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.difgeo.2025.102311
Chi Hin Chan , Magdalena Czubak , Carlos Pinilla Suarez
The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and forms. We further extend the Hodge decomposition to the Sobolev space for general k-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on .
{"title":"Hodge decomposition of the Sobolev space H1 on a space form of nonpositive curvature","authors":"Chi Hin Chan , Magdalena Czubak , Carlos Pinilla Suarez","doi":"10.1016/j.difgeo.2025.102311","DOIUrl":"10.1016/j.difgeo.2025.102311","url":null,"abstract":"<div><div>The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> forms. We further extend the Hodge decomposition to the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> for general <em>k</em>-forms on non-compact manifolds of nonpositive constant sectional curvature. As a result, we also obtain a decomposition on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102311"},"PeriodicalIF":0.7,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-10DOI: 10.1016/j.difgeo.2025.102308
Yalin Sun, Ruiwei Xu
In this paper we classify a kind of special Calabi hypersurfaces with negative constant sectional curvature in Calabi affine geometry. Meanwhile, we find a class of new Euclidean complete and Calabi complete affine hypersurfaces, which satisfy the affine maximal type equation and the Abreu equation with negative constant scalar curvatures.
{"title":"A class of new complete affine maximal type hypersurfaces","authors":"Yalin Sun, Ruiwei Xu","doi":"10.1016/j.difgeo.2025.102308","DOIUrl":"10.1016/j.difgeo.2025.102308","url":null,"abstract":"<div><div>In this paper we classify a kind of special Calabi hypersurfaces with negative constant sectional curvature in Calabi affine geometry. Meanwhile, we find a class of new Euclidean complete and Calabi complete affine hypersurfaces, which satisfy the affine maximal type equation and the Abreu equation with negative constant scalar curvatures.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102308"},"PeriodicalIF":0.7,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-07DOI: 10.1016/j.difgeo.2025.102307
Saajid Chowdhury , Hechen Hu , Matthew Romney , Adam Tsou
We study the properties of surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the condition locally. The main facts about surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of surfaces. We also show that surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most κ. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.
{"title":"On CAT(κ) surfaces","authors":"Saajid Chowdhury , Hechen Hu , Matthew Romney , Adam Tsou","doi":"10.1016/j.difgeo.2025.102307","DOIUrl":"10.1016/j.difgeo.2025.102307","url":null,"abstract":"<div><div>We study the properties of <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> condition locally. The main facts about <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces. We also show that <span><math><mi>CAT</mi><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most <em>κ</em>. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102307"},"PeriodicalIF":0.7,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-28DOI: 10.1016/j.difgeo.2025.102306
Zhaoting Wei
Cohesive modules give a dg-enhancement of the bounded derived category of coherent sheaves on a complex manifold via superconnections. In this paper we discuss the deformation theory of cohesive modules on compact complex manifolds. This generalizes the deformation theory of holomorphic vector bundles and coherent sheaves. We also develop the theory of Kuranishi maps and obstructions of deformations of cohesive modules and give some examples of unobstructed deformations.
{"title":"Deformations of cohesive modules on compact complex manifolds","authors":"Zhaoting Wei","doi":"10.1016/j.difgeo.2025.102306","DOIUrl":"10.1016/j.difgeo.2025.102306","url":null,"abstract":"<div><div>Cohesive modules give a dg-enhancement of the bounded derived category of coherent sheaves on a complex manifold via superconnections. In this paper we discuss the deformation theory of cohesive modules on compact complex manifolds. This generalizes the deformation theory of holomorphic vector bundles and coherent sheaves. We also develop the theory of Kuranishi maps and obstructions of deformations of cohesive modules and give some examples of unobstructed deformations.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102306"},"PeriodicalIF":0.7,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-27DOI: 10.1016/j.difgeo.2025.102303
N.T. Dung , L.G. Linh , P.B. Ngan , A. Upadhyay
In this paper, we revise some results on rigidity and vanishing properties obtained by Cuong et al. in [5] on n-dimensional totally real minimal submanifolds M immersed in complex space forms , for . We extend the range of p in their paper.
{"title":"Rigidity and vanishing results on totally real submanifolds under Lp-integrable conditions","authors":"N.T. Dung , L.G. Linh , P.B. Ngan , A. Upadhyay","doi":"10.1016/j.difgeo.2025.102303","DOIUrl":"10.1016/j.difgeo.2025.102303","url":null,"abstract":"<div><div>In this paper, we revise some results on rigidity and vanishing properties obtained by <em>Cuong</em> et al. in <span><span>[5]</span></span> on <em>n</em>-dimensional totally real minimal submanifolds <em>M</em> immersed in complex space forms <span><math><msup><mrow><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span>, for <span><math><mi>c</mi><mo>≤</mo><mn>0</mn></math></span>. We extend the range of <em>p</em> in their paper.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102303"},"PeriodicalIF":0.7,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-16DOI: 10.1016/j.difgeo.2025.102305
Antonio De Nicola , Ivan Yudin
We introduce the Darboux-Lie derivative along a vector field of fiber bundle maps from natural bundles to associated fiber bundles and study its properties.
引入了天然纤维束映射到伴生纤维束的矢量场上的达布-李导数，并研究了它的性质。
{"title":"Darboux-Lie derivatives","authors":"Antonio De Nicola , Ivan Yudin","doi":"10.1016/j.difgeo.2025.102305","DOIUrl":"10.1016/j.difgeo.2025.102305","url":null,"abstract":"<div><div>We introduce the Darboux-Lie derivative along a vector field of fiber bundle maps from natural bundles to associated fiber bundles and study its properties.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102305"},"PeriodicalIF":0.7,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-15DOI: 10.1016/j.difgeo.2025.102304
Dung Phuong Phan , Tuan Anh Bui , Alexander D. Rahm
This article investigates the torsion homology behaviour in towers of Oeljeklaus–Toma (OT) manifolds. This adapts an idea of Silver–Williams from knot theory to OT manifolds and extends it to higher degree homology groups.
In the case of surfaces, i.e. Inoue surfaces of type , the torsion grows exponentially in both (as was established by Bräunling) and (our result) according to a parameter which already plays a role in Inoue's classical paper, and we obtain that the torsion vanishes in all higher degrees. This motivates our presented machine calculations for OT manifolds of one dimension higher.
{"title":"Studies of the torsion in the homology of Oeljeklaus–Toma manifolds","authors":"Dung Phuong Phan , Tuan Anh Bui , Alexander D. Rahm","doi":"10.1016/j.difgeo.2025.102304","DOIUrl":"10.1016/j.difgeo.2025.102304","url":null,"abstract":"<div><div>This article investigates the torsion homology behaviour in towers of Oeljeklaus–Toma (OT) manifolds. This adapts an idea of Silver–Williams from knot theory to OT manifolds and extends it to higher degree homology groups.</div><div>In the case of surfaces, i.e. Inoue surfaces of type <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>, the torsion grows exponentially in both <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> (as was established by Bräunling) and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (our result) according to a parameter which already plays a role in Inoue's classical paper, and we obtain that the torsion vanishes in all higher degrees. This motivates our presented machine calculations for OT manifolds of one dimension higher.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102304"},"PeriodicalIF":0.7,"publicationDate":"2025-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-14DOI: 10.1016/j.difgeo.2025.102302
Makiko Sumi Tanaka , Hiroyuki Tasaki
In the authors' article in 2020 we classified and explicitly described maximal antipodal sets of Grassmann manifolds, compact symmetric spaces , and their quotient spaces. In this article we show similar results for compact symmetric spaces and and their quotient spaces by using realizations of these as polars in disconnected compact Lie groups.
{"title":"Maximal antipodal sets of compact classical symmetric spaces and their cardinalities. II","authors":"Makiko Sumi Tanaka , Hiroyuki Tasaki","doi":"10.1016/j.difgeo.2025.102302","DOIUrl":"10.1016/j.difgeo.2025.102302","url":null,"abstract":"<div><div>In the authors' article in 2020 we classified and explicitly described maximal antipodal sets of Grassmann manifolds, compact symmetric spaces <span><math><mi>C</mi><mi>I</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, <span><math><mi>D</mi><mi>I</mi><mi>I</mi><mi>I</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and their quotient spaces. In this article we show similar results for compact symmetric spaces <span><math><mi>U</mi><mi>I</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>U</mi><mi>I</mi><mi>I</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and their quotient spaces by using realizations of these as polars in disconnected compact Lie groups.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102302"},"PeriodicalIF":0.7,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-06DOI: 10.1016/j.difgeo.2025.102301
Gregory J. Galloway
We present a version of the Lorentzian splitting theorem under a weakened Ricci curvature condition. The proof makes use of basic properties of achronal limits [19], [20], together with the geometric maximum principle for spacelike hypersurfaces in [1]. Our version strengthens a related result in [29] in the globally hyperbolic setting by removing a certain boundedness condition on the Ricci curvature.
{"title":"A note on the Lorentzian splitting theorem","authors":"Gregory J. Galloway","doi":"10.1016/j.difgeo.2025.102301","DOIUrl":"10.1016/j.difgeo.2025.102301","url":null,"abstract":"<div><div>We present a version of the Lorentzian splitting theorem under a weakened Ricci curvature condition. The proof makes use of basic properties of achronal limits <span><span>[19]</span></span>, <span><span>[20]</span></span>, together with the geometric maximum principle for <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> spacelike hypersurfaces in <span><span>[1]</span></span>. Our version strengthens a related result in <span><span>[29]</span></span> in the globally hyperbolic setting by removing a certain boundedness condition on the Ricci curvature.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"101 ","pages":"Article 102301"},"PeriodicalIF":0.7,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145266332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号